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An introduction to diffusion

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The rate of transfer of diffusing substance through unit area of a section is ... of energy to a volume, e.g. a parallelepiped of lengths 2x, 2y, and 2z. ... – PowerPoint PPT presentation

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Title: An introduction to diffusion


1
An introduction to diffusion
  • Thermochron, Fall 2005

2
TEXTS
  • Heat conduction - Carslaw and Jaegger
  • Diffusion - Crank, 1975

3
Diffusion is analogous to heat conduction
  • The rate of transfer of diffusing substance
    through unit area of a section is proportional to
    the concentration gradient measured normal to the
    section (Ficks law, 1855).

4
Dashed for heat sources and sinks
5
Similar to heat transfer- same thing, really
  • Rate of transfer of heat per unit area is
    proportional to the thermal gradient (Fouriers
    Law of heat conduction, 1822).
  • One deals with the diffusion of heat the other
    with the diffusion of mass.

6
Diffusion coefficient
QUESTIONS?? What are the units for D? Why the
negative sign for both forms of diffusion?
7
Answers
  • The minus sign reflects the fact that diffusion
    takes place in the direction opposite to
    increasing concentration (or heat)
  • Units are area/time, e.g. cm2/sec.
  • These simple formulations apply only to perfectly
    isotropic mediums.

8
Deriving the fundamental equation for diffusion
in an isotropic medium
Requires applying the law of conservation of
energy to a volume, e.g. a parallelepiped of
lengths 2x, 2y, and 2z. EinEgeneratedEstoredEo
ut
9
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10
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11
The equivalent equation for heat conduction
12
Some implications
  • The changes in concentration due to diffusion
    over a characteristic time interval t will
    propagate a distance on the order of (Dt)1/2
  • Similar equation for conduction
  • This simplification x (Dt)1/2 gives us a mean to
    determine back of the envelope diffusion scales.
  • If the length scale is x (grain size), how long
    will it take for diffusion to operate at that
    length scale?

Note As you will see soon, diffusion is a
thermally activate process and the assumption of
constant D is a very simplifying one. However, D
is typically constant over a given small range of
temperatures.
13
Example
  • Upper mantle minerals have grain sizes of mm
    length scale. Lets say 1 mm on average.
  • The diffusion coefficient for most elements in
    olivines and pyroxenes (at T 1000 0C) is on the
    order of 1 x 10-15 cm2/sec.
  • What is the time scale of diffusional
    equilibration of a mantle rock and what is the
    prospect of using these rocks for geochronology
    of mantle events?

14
Answers
  • Takes less than 1 My (in this case 0.3 Ma) to
    erase any diffusional gradients
  • Prospects of mantle rocks to preserve any kind of
    diffusional gradients and thus age info not
    good to zero.

15
Same applies to heat conduction
  • Similar to x (Dt)1/2 , one can apply x (Kt)1/2
    to heat conduction
  • E.g. what is the time scale for a granite body of
    size x to cool?

r
cold
1 km body, 10 km, 100 km body
hot
Note the non-linearity
16
  • Does a magma body ten times larger cool ten time
    slower?

100 times slower !
17
Goals
  • Fundamentally, we are in pursuit of diffusion
    laws and the solutions to those laws because they
    constrain the effective temperatures below which
    the effective transport of elements and isotopes
    of interest stops.
  • That is when the isotopic clock starts and
    quantifying the conditions under which that
    happens is equally important as the decay process
    itself.

18
A road map for the next three lectures
  • Find simple analogies to exemplify diffusion of
    mass
  • Try to work out analytical solutions to super
    simplified cases (sphere, plane geometries)
  • Bring in the math baggage associated with this
  • Apply these solutions to the classic formulation
    of closure temperature (Dodson, 1973)
  • Modern variations on the closure temperature
    theory

19
An example
  • Imaginary experiment pool of water and drop a
    chlorine tablet. The pool is an infinite
    reservoir that has no chlorine to start with.
  • What will be the distribution of Cl as a function
    of x and t in the pool?????

20
Set initial and boundary conditions
CtabletC0 Cpool0 t0
Differentiating, one obtains the following
solution
21
If the concentration is originally located in a
point source and the medium is a plane
Which is the classic, analytical solution to a
point source and works ok for your pool although
the chlorine tablet is not a infinitely small and
the pool is not a infinitely large sheet..
22
Graphically..
23
  • The problem is symmetric with respect to x
  • The Pi-form of the constant is determined by the
    fact that the chlorine diffuses not just in the
    directions x and -x, but radially around the
    tablet.

24
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25
Analogy
  • A garnet immersed in an infinitely large
    reservoir of biotite

26
Case 2 two long metal bars placed in contact end
to end.
  • Another classic example in the diffusion book.
  • Has an analytical solution
  • Exemplifies the general form of solutions to more
    complicated problems

27
Two infinite half spaces
28
Solution Consider that the half space is
composed of an infinite number of point sources
(our previous problem). Because of that, one can
obtain an analytical solution by superposing the
infinite of point sources. Overall, in math,
this sumation of linear solutions leads to an
exponential distribution of the solution.
29
Looks like the solution is of the form of a
well-known mathematical function called the error
function (erf)
Solutions to erf are available in tabulated form
or as an excel add-in etc etc.
30
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31
Solution
This is very similar to the heat conduction
solution for equivalent boundary / initial
conditions which the temperature at the interface
stays ct as the average between the two ts.
applies to dike intrusions etc
32
Solution
33
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34
Case 3 plane source of limited extent -
  • A particular case of 2 with the source is limited
    from -hltxgth
  • Integration is thus from x- h to xh instead of x
    to infinity

35
Solution
36
Some concluding remarks from this intro lecture
  • Mass diffusion is (almost) identical in its
    treatment with heat conduction - aka heat
    diffusion
  • Simplifying calculations can give us order of mag
    info on length and time scales of diffusion
  • Analytical solutions are found for some of the
    simplest initial and boundary conditions
  • The most common type of solution of these
    problems involves the error function
  • More complicated diffusion problems do not have
    analytical solutions discussions so far focused
    on constant D and isotropic 1D examples.
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